Abstract
We study equi-Baire 1 families of functions between Polish spaces X and Y. We show that the respective ε-gauge in the definition of such a family can be chosen upper semi-continuous. We prove that a pointwise convergent sequence of continuous functions forms an equi-Baire 1 family. We study families of separately equi-Baire 1 functions of two variables and show that the family of all sections of separately continuous functions also forms an equi-Baire 1 family. We characterize equi-Baire 1 families of characteristic functions. This leads us to an example witnessing that the exact counterpart of the Arzelà-Ascoli theorem for families of real-valued Baire 1 functions on [0,1] is false. Also, the weak counterpart of this theorem dealing with restrictions to nonmeager sets is false. On the other hand, we obtain a simple proof of the Arzelà-Ascoli type theorem for sequences of equi-Baire 1 real-valued functions in the case of pointwise convergence.
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